Hamiltonian cycles in $k$-partite graphs

TL;DR

This paper establishes an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary $k$-partite graphs with parts up to half the total vertices, extending previous results on balanced graphs.

Contribution

It introduces a general theorem that simplifies checking for robust expansion and applies it to characterize Hamiltonicity in $k$-partite graphs with minimal degree conditions.

Findings

Provides an asymptotically tight degree threshold for Hamiltonian cycles.

Introduces a new structural result for robust expanders.

Characterizes Hamiltonicity in $k$-partite graphs with parts up to size $n/2$.

Abstract

Chen, Faudree, Gould, Jacobson, and Lesniak determined the minimum degree threshold for which a balanced $k$-partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary $k$-partite graphs in which all parts have at most $n/2$ vertices (a necessary condition). To do this, we first prove a general result which both simplifies the process of checking whether a graph $G$ is a robust expander and gives useful structural information in the case when $G$ is not a robust expander. Then we use this result to prove that any $k$-partite graph satisfying the minimum degree condition is either a robust expander or else contains a Hamiltonian cycle directly.